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## PowerPoint Slideshow about ' The Empirical FT .' - tashya-stevenson

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What is the large sample distribution of the EFT?

Evaluate first and second-order cumulants

Bound higher cumulants

Normal is determined by its moments

Already used to study rate estimate

Tapering makes

Get asymp independence for different frequencies

The frequencies 2r/T are special, e.g. T(2r/T)=0, r 0

Also get asymp independence if consider separate stretches

p-vector version involves p by p spectral density matrix fXX( )

Estimation of the (power) spectrum.

An estimate whose limit is a random variable

The estimate is asymptotically unbiased

Final term drops out if = 2r/T 0

Best to correct for mean, work with

Periodogram values are asymptotically independent since dT values are -

independent exponentials

Use to form estimates

Estimation of finite dimensional .

approximate likelihood (assuming IT values independent exponentials)

Smoothed periodogram.

Advantages of frequency domain approach.

techniques formany stationary processes look the same

approximate i.i.d sample values

assessing models (character of departure)

time varying variant

...

partial spectra - trivariate (M,N,O)

Remove linear time invariant effect of M from N and O and examine coherency of residuals,

An operation carrying a point process, M, into another, N

N = S[M]

S = {input,mapping, output}

Pr{dN(t)=1|M} = { + a(t-u)dM(u)}dt

System identification: determining the charcteristics of a system from inputs and corresponding outputs

{M(t),N(t);0 t<T}

Like regression vs. bivariate

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